Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:quant-ph/0108055 · Journal of Physics A Mathematical and General

A canonical approach to the quantization of the damped harmonic oscillator

Rabin Banerjee, Pradip Mukherjee

Published Jun 28, 2002• 46 citations
arXivHTMLPDF

Abstract

We provide a new canonical approach for studying the quantum mechanical damped harmonic oscillator based on the doubling of degrees of freedom approach. Explicit expressions for Lagrangians of the elementary modes of the problem, characterizing both forward and backward time propagation, are given. A Hamiltonian analysis, showing the equivalence with the Lagrangian approach, is also done. Based on this Hamiltonian analysis, the quantization of the model is discussed.

DOI
10.1088/0305-4470/35/27/305

Related papers

Quantization of Damped Mechanical Systems Based on the Doubling of Degrees of Freedom

Khaled I. Nawafleh

Quantizing dissipative systems like damped harmonic oscillators remains an ongoing challenge in theoretical physics due to their non-conservative nature.This study addresses this issue by employing the method of doubling the degrees of freedom, transforming an open, dissipative system into a closed, conservative one through the introduction of a time-reversed mirror system.The aim is to derive a quantum model consistent with classical mechanics using the Hamilton-Jacobi formalism and to formulate an equivalent Schrödinger equation.Through canonical transformations and analytical solutions, we demonstrate that damping effectively vanishes in the quantum model.These results offer new insights into the quantization of non-conservative systems and contribute to the advancement of quantum open-system theory.

Science International•May 19, 2025•0 citations

A new approach to the quantization of the damped harmonic oscillator

Matthew J. Blacker, David L. Tilbrook

In this paper, a new approach for constructing Lagrangians for driven and undriven linearly damped systems is proposed, by introducing a redefined time coordinate and an associated coordinate transformation to ensure that the resulting Lagrangian satisfies the Helmholtz conditions. The approach is applied to canonically quantize the damped harmonic oscillator and although it predicts an energy spectrum that decays at the same rate to previous models, unlike those approaches it recovers the classical critical damping condition, which determines transitions between energy eigenstates, and is therefore consistent with the correspondence principle. It is also demonstrated how to apply the procedure to a driven damped harmonic oscillator.

arXiv:2107.05827•Jul 13, 2021•0 citations

Canonical transformation on an open system

Qiankai Yao, Bing Ma, Xue-Chao Feng +2 more

A time-dependent Lagrangian is constructed. This time-dependent Lagrangian can help us to quantize the damped system in the frameworks of canonical quantization, and solve the non-autonomous quantum problem in different representations. We evaluate the measuring effect on the quantum system and find a linear canonical transformation between the original system and the measured one. Finally, the damped harmonic oscillator is discussed.

Il Nuovo Cimento B•Jan 1, 2004•0 citations

Alternative approach to the quantization of the damped harmonic oscillator

Matthew J. Blacker, David L. Tilbrook

In this paper, an alternative approach for constructing Lagrangians for driven and undriven linearly damped systems is proposed, by introducing a redefined time coordinate and an associated coordinate transformation to ensure that the resulting Lagrangian satisfies the Helmholtz conditions. The approach is applied to canonically quantize the damped harmonic oscillator and although it predicts an energy spectrum that decays at the same rate to previous models, unlike those approaches it recovers the classical critical damping condition, which determines transitions between energy eigenstates, and is therefore consistent with the correspondence principle. It is also demonstrated how to apply the procedure to a driven damped harmonic oscillator.

Physical Review A•Sep 13, 2021•10 citations

Quantization of the damped harmonic oscillator

Mohammed Serhan, M. Abusini, Ahmed Al-Jamel +2 more

In this work, a suitable Hamiltonian that describes the damped harmonic oscillator is constructed. Besides, the Hamilton-Jacobi equation for this dissipative system is written and the action function is obtained. Then, the system is quantized using two methods: the WKB approximation and the canonical quantization. In addition, evolution of the probability density is plotted at different times, and it is observed that the width of these densities decreases with time. The average energy of the system is also calculated and found to decrease with time.

Journal of Mathematical Physics•Aug 1, 2018•18 citations

Comment on “Quantization of the damped harmonic oscillator” [J. Math. Phys. 59, 082105 (2018)]

Francisco Marcelo Fernández

Serhan et al. [J. Math. Phys. 59, 082105 (2018)] derived a quantum-mechanical Hamiltonian operator from the classical equations of motion for the damped harmonic oscillator and obtained its eigenvalues and eigenfunctions by canonical quantization. They claim that the eigenvalues are real, but here we show that they are in fact complex.

Journal of Mathematical Physics•Sep 1, 2019•2 citations
We provide a new canonical approach for studying the quantum mechanical damped harmonic oscillator based on the doubling of degrees of freedom approach. Explicit expressions for Lagrangians of the elementary modes of the problem, characterizing both forward and backward time propagation, are given. A Hamiltonian analysis, showing the equivalence with the Lagrangian approach, is also done. Based on this Hamiltonian analysis, the quantization of the model is discussed.
Quantizing dissipative systems like damped harmonic oscillators remains an ongoing challenge in theoretical physics due to their non-conservative nature.This study addresses this issue by employing the method of doubling the degrees of freedom, transforming an open, dissipative system into a closed, conservative one through the introduction of a time-reversed mirror system.The aim is to derive a quantum model consistent with classical mechanics using the Hamilton-Jacobi formalism and to formulate an equivalent Schrödinger equation.Through canonical transformations and analytical solutions, we demonstrate that damping effectively vanishes in the quantum model.These results offer new insights into the quantization of non-conservative systems and contribute to the advancement of quantum open-system theory.
In this paper, a new approach for constructing Lagrangians for driven and undriven linearly damped systems is proposed, by introducing a redefined time coordinate and an associated coordinate transformation to ensure that the resulting Lagrangian satisfies the Helmholtz conditions. The approach is applied to canonically quantize the damped harmonic oscillator and although it predicts an energy spectrum that decays at the same rate to previous models, unlike those approaches it recovers the classical critical damping condition, which determines transitions between energy eigenstates, and is therefore consistent with the correspondence principle. It is also demonstrated how to apply the procedure to a driven damped harmonic oscillator.
In this paper, an alternative approach for constructing Lagrangians for driven and undriven linearly damped systems is proposed, by introducing a redefined time coordinate and an associated coordinate transformation to ensure that the resulting Lagrangian satisfies the Helmholtz conditions. The approach is applied to canonically quantize the damped harmonic oscillator and although it predicts an energy spectrum that decays at the same rate to previous models, unlike those approaches it recovers the classical critical damping condition, which determines transitions between energy eigenstates, and is therefore consistent with the correspondence principle. It is also demonstrated how to apply the procedure to a driven damped harmonic oscillator.
In this work, a suitable Hamiltonian that describes the damped harmonic oscillator is constructed. Besides, the Hamilton-Jacobi equation for this dissipative system is written and the action function is obtained. Then, the system is quantized using two methods: the WKB approximation and the canonical quantization. In addition, evolution of the probability density is plotted at different times, and it is observed that the width of these densities decreases with time. The average energy of the system is also calculated and found to decrease with time.